Optimal quadrature for haar wavelet spaces

Stefan Heinrich; Fred J. Hickernell; Rong Xian Yue

doi:10.1090/S0025-5718-03-01531-X

Optimal quadrature for haar wavelet spaces

Stefan Heinrich^*
, Fred J. Hickernell
, Rong Xian Yue

^*Corresponding author for this work

Research output: Contribution to journal › Journal article › peer-review

23 Citations (Scopus)

Abstract

This article considers the error of the scrambled equidistribution quadrature rules in the worst-case, random-case, and average-case settings. The underlying space of integrands is a Hilbert space of multidimensional Haar wavelet series, ℋ_wav. The asymptotic orders of the errors are derived for the case of the scrambled (λ, t, m, s)-nets and (t, s)-sequences. These rules are shown to have the best asymptotic convergence rates for any random quadrature rule for the space of integrands ℋ _wav.

Original language	English
Pages (from-to)	259-277
Number of pages	19
Journal	Mathematics of Computation
Volume	73
Issue number	245
DOIs	https://doi.org/10.1090/S0025-5718-03-01531-X
Publication status	Published - Jan 2004
Externally published	Yes

User-Defined Keywords

High dimensional integration
Lower bounds
Monte Carlo methods
Quasi-Monte Carlo methods

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10.1090/S0025-5718-03-01531-X

Cite this

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title = "Optimal quadrature for haar wavelet spaces",

abstract = "This article considers the error of the scrambled equidistribution quadrature rules in the worst-case, random-case, and average-case settings. The underlying space of integrands is a Hilbert space of multidimensional Haar wavelet series, ℋwav. The asymptotic orders of the errors are derived for the case of the scrambled (λ, t, m, s)-nets and (t, s)-sequences. These rules are shown to have the best asymptotic convergence rates for any random quadrature rule for the space of integrands ℋ wav.",

keywords = "High dimensional integration, Lower bounds, Monte Carlo methods, Quasi-Monte Carlo methods",

author = "Stefan Heinrich and Hickernell, \{Fred J.\} and Yue, \{Rong Xian\}",

note = "Funding information: This work was partially supported by a Hong Kong Research Grants Council grant HKBU/2030/99P, by Hong Kong Baptist University grant FRG/97-98/II-99, by Shanghai NSF Grant 00JC14057, and by Shanghai Higher Education STF grant 01D01-1. Publisher copyright: {\textcopyright} 2003 American Mathematical Society",

year = "2004",

month = jan,

doi = "10.1090/S0025-5718-03-01531-X",

language = "English",

volume = "73",

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publisher = "American Mathematical Society",

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T1 - Optimal quadrature for haar wavelet spaces

AU - Heinrich, Stefan

AU - Hickernell, Fred J.

AU - Yue, Rong Xian

N1 - Funding information: This work was partially supported by a Hong Kong Research Grants Council grant HKBU/2030/99P, by Hong Kong Baptist University grant FRG/97-98/II-99, by Shanghai NSF Grant 00JC14057, and by Shanghai Higher Education STF grant 01D01-1. Publisher copyright: © 2003 American Mathematical Society

PY - 2004/1

Y1 - 2004/1

N2 - This article considers the error of the scrambled equidistribution quadrature rules in the worst-case, random-case, and average-case settings. The underlying space of integrands is a Hilbert space of multidimensional Haar wavelet series, ℋwav. The asymptotic orders of the errors are derived for the case of the scrambled (λ, t, m, s)-nets and (t, s)-sequences. These rules are shown to have the best asymptotic convergence rates for any random quadrature rule for the space of integrands ℋ wav.

AB - This article considers the error of the scrambled equidistribution quadrature rules in the worst-case, random-case, and average-case settings. The underlying space of integrands is a Hilbert space of multidimensional Haar wavelet series, ℋwav. The asymptotic orders of the errors are derived for the case of the scrambled (λ, t, m, s)-nets and (t, s)-sequences. These rules are shown to have the best asymptotic convergence rates for any random quadrature rule for the space of integrands ℋ wav.

KW - High dimensional integration

KW - Lower bounds

KW - Monte Carlo methods

KW - Quasi-Monte Carlo methods

UR - https://www.jstor.org/stable/4099869

UR - https://www.scopus.com/pages/publications/0942278836

U2 - 10.1090/S0025-5718-03-01531-X

DO - 10.1090/S0025-5718-03-01531-X

M3 - Journal article

AN - SCOPUS:0942278836

SN - 0025-5718

VL - 73

SP - 259

EP - 277

JO - Mathematics of Computation

JF - Mathematics of Computation

IS - 245

ER -

Optimal quadrature for haar wavelet spaces

Abstract

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